When I read a book of set theory written by Charles Pinter, in chapter 1 section 7, the author says
If $A$ is any set, there is an element $a\in A$ such that $a\cap A=\emptyset$.
The author calls it the axiom of foundation. Let $A=\{red,blue\}$, by this axiom, we know $red \cap A=\emptyset$, but in native set theory $red \cap A$ makes no sense, I know in axiomatic set theory everything is set, but how to explain this contradiction? Are set in naive set theory and set in axiomatic set theory the same?
There are several tensions between naive and axiomatic set theory. The big one, of course, is "naive" versus "axiomatic" - are we looking at an informal or formal theory? This question, though, focuses on a different tension: the tension between being maximally inclusive and being meaningfully specific.
In short, the situation is this:
Here's a bit more detail:
Things like "red" and "blue" simply don't exist from the point of view of axiomatic set theory (at least, the usual version; not every form of axiomatic set theory includes the axiom of foundation, or even asserts that all objects are sets!). If you wish, you can think of standard axiomatic set theory as describing a portion of the mathematical universe - namely, the part which can be "built out of $\emptyset$" just by the basic set operations (including powerset and transfinite recursion - so, maybe "basic" is misleading!).
Put another way, it's obvious that any theory of sets needs to be able to talk about $\emptyset$. Well, $\{\emptyset\}$ is also a set, so that should be included. By contrast, if all we want to do is talk about sets, there's no obvious reason why something like "red" should be in our universe. Axiomatic set theory studies the concept of set on its own in a sense. In light of this, it should be surprising that something actually interesting (let alone powerful enough to implement all of mathematics) comes out!